Model reduction of parametric ordinary differential equations via autoencoders: representation properties and convergence analysis
For researchers in scientific computing and model reduction, this work provides a convergence analysis and representation properties for autoencoder-based reduction, though the approach is incremental over existing nonlinear reduction methods.
The paper proposes a reduced-order modeling approach for nonlinear parametric ODEs using autoencoders for dimensionality reduction, demonstrating robustness and accuracy in complex nonlinear settings without sacrificing accuracy.
We propose a reduced-order modeling approach for nonlinear, parameter-dependent ordinary differential equations (ODE). Dimensionality reduction is achieved using nonlinear maps represented by autoencoders. The resulting low-dimensional ODE is then solved using standard integration in time schemes, and the high-dimensional solution is reconstructed from the low-dimensional one. We investigate the architecture of neural networks for constructing effective autoencoders that hold necessary properties to reconstruct the input manifold with exact representation capabilities. We study the convergence of the reduced-order model to the high-fidelity one. Numerical experiments show the robustness and accuracy of our approach in different scenarios, highlighting its effectiveness in highly complex and nonlinear settings without sacrificing accuracy. Moreover, we examine how the reduction influences the stability properties of the reconstructed high-dimensional solution.